An alternative definition of ordered pairs. This definition removes a
hypothesis from its defining theorem (see altopth25542), making it more
convenient in some circumstances. (Contributed by Scott Fenton,
22-Mar-2012.)

Two alternate ordered pairs are equal iff the singletons of their
respective elements are equal. Note that this holds regardless of sethood
of any of the elements. (Contributed by Scott Fenton, 16-Apr-2012.)

The alternate ordered pair theorem. If two alternate ordered pairs are
equal, their first elements are equal and their second elements are
equal. Note that and are
not required to be a set due to a
peculiarity of our specific ordered pair definition, as opposed to the
regular ordered pairs used here, which (as in opth4390), requires
to be a set. (Contributed by Scott Fenton, 23-Mar-2012.)

Membership in a Euclidean space. We define Euclidean space here using
Cartesian coordinates over space. We later abstract away from
this using Tarski's geometry axioms, so this exact definition is
unimportant. (Contributed by Scott Fenton, 3-Jun-2013.)

The binary relationship form of the betweenness predicate. The
statement should be informally read as "
lies on a line segment between and . This exact definition
is abstracted away by Tarski's geometry axioms later on. (Contributed
by Scott Fenton, 3-Jun-2013.)

The binary relationship form of the congruence predicate. The statement
Cgr should be read informally as "the
dimensional point is as far from as is from
, or
"the line segment is
congruent to the line segment .
This particular definition is encapsulated by Tarski's axioms later on.
(Contributed by Scott Fenton, 3-Jun-2013.)

The unique dimensional axiom. If a point is in dimensional space
and in
dimensional space, then . This axiom is not
traditionally presented with Tarski's axioms, but we require it here as we
are considering spaces in arbitrary dimensions. (Contributed by Scott
Fenton, 24-Sep-2013.)